Pseudorandom sets in Grassmann graph have near-perfect expansion

In several previous works the construction of a computationally hard function with respect to a certain class of algorithms or Boolean circuits has been used to derive small pseudo-random spaces. In this paper, we revert this connection by presenting two new direct relations between the efficient construction of pseudo-random (both two-sided and one-sided) sets for Boolean affine spaces and the explicit construction of Boolean functions having hard branching program complexityIn the case of 1-read branching programs (1-Br.Pr.), we show that the construction of non trivial (i.e. of cardinality 2o(n)) discrepancy sets (i.e. two-sided pseudo-random sets) for Boolean affine spaces of dimension greater than n/2 yield a set of explicit Boolean functions having very hard 1-Br.Pr. size. By combining the best known construction of ∈-biased sample spaces for linear tests and a simple “Reduction” Lemma, we derive the required discrepancy set and obtain a Boolean function in P having 1-Br.Pr. size not smaller than \(2^{n - O\left( {log^2 n} \right)} \) and a Boolean function in DTIME\(\left( {2^{O\left( {log^2 n} \right)} } \right)\) having 1-Br.Pr. size not smaller than 2n-O(log n). The latter bound is optimal and both of them are exponential improvements over the best previously known lower bound that was \(2^{n - 3n^{1/2} } \)[21].As for non deterministic syntactic k-read branching programs (k-Br.Pr.), we introduce a new method to derive explicit, exponential lower bounds that involves the construction of hitting sets (one-sided pseudo-random sets) for affine spaces of dimension o(n=2). Using an appropriate “orthogonal” representation of small Boolean affine spaces, we efficiently construct these hitting sets thus obtaining an explicit Boolean function in P that has k-Br.Pr. size not smaller than \(2^{n^{1 - o\left( 1 \right)} } \) for any \(k = o\left( {\frac{{\log n}}{{\log \log n}}} \right)\). This improves over the previous best known lower bounds given in [8,11,17] for some range of k

Let $\mathbb{F}_q$ denote a finite field with $q$ elements. Let $n,k$ denote integers with $n>2k\geq 6$. Let $V$ denote a vector space over $\mathbb{F}_{q}$ that has dimension $n$. The vertex set of the Grassmann graph $J_q(n,k)$ consists of the $k$-dimensional subspaces of $V$. Two vertices of $J_q(n,k)$ are adjacent whenever their intersection has dimension $k-1$. Let $\partial$ denote the path-length distance function of $J_q(n,k)$. Pick vertices $x,y$ of $J_q(n,k)$ such that $1<\partial(x,y)<k$. Let $\text{Stab}(x,y)$ denote the subgroup of $GL(V)$ that stabilizes both $x$ and $y$. In this paper, we investigate the orbits of $\text{Stab}(x,y)$ acting on the local graph $\Gamma(x)$. We show that there are five orbits. By construction, these five orbits give an equitable partition of $\Gamma(x)$; we find the corresponding structure constants. In order to describe the five orbits more deeply, we bring in a Euclidean representation of $J_q(n,k)$ associated with the second largest eigenvalue of $J_q(n,k)$. By construction, for each orbit its characteristic vector is represented by a vector in the associated Euclidean space. We compute many inner products and linear dependencies involving the five representing vectors.

This paper presents a self-stabilizing algorithm for computing multiple disjoint maximal independent sets in a graph. The first set is a maximal independent set in the entire graph, the second set is a maximal independent set in the subgraph formed by removing the nodes in the first independent set, and each subsequent set is computed similarly by excluding the nodes from all previous sets. The algorithm introduces a novel method by using a single local integer variable for each node to determine its membership in a specific independent set, offering a more efficient solution compared to earlier methods. Additionally, this algorithm provides an upper bound on the chromatic number required for the graph coloring problem. The solution is developed under the central daemon model and guarantees termination after a finite number of moves.

A parallel algorithm for finding a near-maximum independent set in a circle graph is presented. An independent set in a graph is a set of vertices, no two of which are adjacent. A maximum independent set is an independent set whose cardinality is the largest among all independent sets of a graph. The algorithm is modified for predicting the secondary structure in ribonucleic acids (RNA). The proposed system, composed of an n neural network array (where n is the number of edges in the circle graph of the number of possible base pairs), not only generates a near-maximum independent set but also predicts the secondary structure of ribonucleic acids within several hundred iteration steps. The simulator discovered several solutions which are more stable structures, in a sequence of 359 bases from the potato spindle tuber viroid, than previously proposed structures.

Let $G$ be a graph. Then a subset $C$ of vertices of $G$ is called a vertex cover hop dominating if $C$ is both a vertex cover and a hop dominating of $G$. The vertex cover hop domination number of $G$, denoted by $\gamma_{vch}(G)$, is the minimum cardinality among all vertex cover hop dominating sets in $G$. In this paper, we initiate the study of vertex cover hop domination in a graph and we determine its relations with other parameters in graph theory. We characterize the vertex cover hop dominating sets in some special graphs, join, and corona of two graphs and we finally obtain the exact values or bounds of the parameters of these graphs.

Finding maximum independent sets in graphs with bounded maximum degree is a well-studied NP-complete problem. We study two approaches for finding approximate solutions, and obtain several improved performance ratios.The first is a subgraph removal schema introduced in our previous paper. Using better component algorithms, we obtain an efficient method with a Δ/6(1+o(1)) performance ratio. We then produce an implementation of a theorem of Ajtai et al. on the independence number of clique-free graphs, and use it to obtain a O(Δ/loglogΔ) performance ratio with our schema. This is the first o(Δ) ratio.The second is a local search method of Berman and Fürer for which they proved a fine performance ratio but by using extreme amounts of time. We show how to substantially decrease the computing requirements while maintaining the same performance ratios of roughly (Δ+3)/5 for graphs with maximum degree Δ. We then show that a scaled-down version of their algorithm yields a (Δ+3)/4 performance, improving on previous bounds for reasonably efficient methods.KeywordsLocal SearchMaximum DegreePerformance RatioFree GraphIndependence NumberThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

An independent set S of vertices in a graph G is an independent position set if no three vertices of S lie on a common geodesic. An independent position set of maximum size is an ip-set of G. The cardinality of an ip-set is the independent position number, denoted by ip(G). In this paper, we introduce and study the independent position number of a graph. Certain general properties of these concepts are discussed. Graphs of order n having the independent position number 1 or n − 1 are characterized. Bounds for the independent position number of Cartesian and Lexicographic product graphs are determined and the exact value for Corona product graphs are obtained. Finally, some realization results are proved to show that there is no general relationship between independent position sets and other related graph invariants

A [Formula: see text]-design over regular graph [Formula: see text] of degree [Formula: see text] is an ordered pair [Formula: see text], where [Formula: see text] and [Formula: see text] is the set of geodetic sets in [Formula: see text] called blocks such that two vertices [Formula: see text] and [Formula: see text] which are [Formula: see text] associates occur together in [Formula: see text] blocks, the numbers [Formula: see text] being independent of the choice of the pair [Formula: see text] and [Formula: see text]. In this paper, we obtain Partially Balanced Incomplete Block (PBIB)-designs arising from geodetic sets in some classes of graphs. Also, we give a complete list of PBIB-designs with respect to the geodetic sets for cubic graphs of order at most [Formula: see text]. The discussion of non-existence of some designs corresponding to geodetic sets from certain graphs concludes the paper.

Problems in the Ramsey theory form a well-known family of mathematical problems. We present the exact value of the Ramsey number R(θ6, W5), where θ6 is the set of theta graphs of order 6 and W5 is the wheel graph of order 5. This result solves the challenging problem of finding exact values of R(θn, W5) for every n ≥ 5.

Maximal and maximum dissociation sets in general and triangle-free graphs

In this paper, we develop efficient exact and approximate algorithms for computing a maximum independent set in random graphs. In a random graph G, each pair of vertices are joined by an edge with a probability p, where p is a constant between 0 and 1. We show that a maximum independent set in a random graph that contains n vertices can be computed in expected computation time . In addition, we show that, with high probability, the parameterized independent set problem is fixed parameter tractable in random graphs and the maximum independent set in a random graph in n vertices can be approximated within a ratio of in expected polynomial time.

Attainable estimates of the number of independent sets in graphs with a given size of the maximal independent set are obtained. Three graph classes—trees, forests, and the class of all graphs—are considered. Extremal graphs are described.

Counting independent sets in a tolerance graph

Maximal independent sets in caterpillar graphs

Independent set of intersection graphs of convex objects in 2D